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Branch Decompositions

ANDBranch width



Introduction

The notions of branch decompositions and treedecompositions and their respective connectivityinvariants, branch width and tree width, are twoemerging techniques for discrete optimization that alsoencompass the fields of graph theory, computer science,and operations research.

The origins of branch width and tree width are deeplyrooted in the proof of the Graph Minors Theorem,formally known as Wagner’s conjecture.

The algorithmic importance of the branch decompositionand tree decomposition was not realized until Courcelleand Arnborg et al. showed that several NP-hard problemsposed in monadic second-order logic can be solved inpolynomial time using dynamic programming techniqueson input graphs with bounded tree width or Branch

width.



A problem which is NP-hard implies that as long as it isnot proven that P = NP we cannot expect to have apolynomial time algorithm for the problem. Thesetechniques are referred to as tree decomposition based

algorithms and branch decomposition based algorithms,respectively.

Branch decomposition and tree decomposition basedalgorithms are important in discrete optimization becausethey have been shown to be effective for combinatorial

optimization problems like the ring-routing problem, thetraveling salesman problem, frequency assignment,general minor containment, and the optimal branchdecomposition problem.



The procedure to solve an optimization problem withbounded branch width or tree width involves two steps:

(i) computation of a (good) branch/tree decomposition

(ii) application of an algorithm that solves instances ofbounded branch width/tree width in polynomial time.

Since the branch width or tree width is considered to be aconstant, not part of the input, this value may occur in theexponent of the complexity of both running time andspace requirements. Hence, it is important to have adecomposition of width as small as possible. The problemof minimizing this quantity is however NP-hard itself.



Branch Decompositions

Let G = (V,E) be a hypergraph and T be aternary tree with |E(G)| leaves. Let ν be a

bijection from the edges of G to the leavesof T . Then the pair (T, ν) is called a branchdecomposition of G.

A partial branch decomposition is a branchdecomposition without the restriction ofevery non-leaf node having degree 3



Some definitions

A separation of a graph G is a pair (G1,G2) ofsubgraphs with G1 ∪G2 = G and E(G1 ∩G2)= ∅,and the order of this separation is defined as

|V (G1 ∩G2)|. Let (T, ν) be a branch decomposition. Then

removing an edge, say e, from T partitions theedges of G into two subsets Ae and Be. The middleset of e, denoted mid(e), is the set of vertices of Gthat are incident to the edges in Ae and the edgesin Be and the width of an edge e, denoted|mid(e)|, is the order of the separation(G[Ae],G[Be]).



The width of a branch decomposition (T, ν)is the maximum width among all edges ofthe decomposition.

The branch width of G, denoted by β(G), isthe minimum width over all branchdecompositions of G.

A branch decomposition of G with widthequal to the branch width is an optimalbranch decomposition of G



Tree Decompositions

The notions of a tree decomposition and tree width wereintroduced by Robertson and Seymour; they measure thetree-likeness of a graph.

Given a graphG

= (V

,E), a tree decomposition is a pair(X , T), where X = {X1, ..., Xn} is a family of subsets of V ,

and T is a tree whose nodes are the subsets Xi, satisfyingthe following properties:

The union of all sets Xi equals V . That is, each graphvertex is associated with at least one tree node.

For every edge (v, w) in the graph, there is a subset Xi thatcontains both v and w . That is, vertices are adjacent in thegraph only when the corresponding subtrees have a node

in common.



First-order logic FO and monadic second-order

logic MSO

We assume that we have an infinite supply of individual

variables, usually denoted by the lowercase letters x,y, z,and an infinite supply of set variables, usually denoted by

uppercase lettersX

,Y

,Z

.First-order formulas

in thelanguage of graphs are built up from atomic formulasE(x,y) and x = y by using the usual Boolean connectives￢ (negation), ∧ (conjunction), ∨ (disjunction), → (implication), and ↔ (bi-implication) and existential

quantification ∃x and universal quantification ∀x overindividual variables. Individual variables range oververtices of a graph.



The atomic formula E(x,y) expresses adjacency, and theformula x = y expresses equality. First-order formulasover labeled graphs may contain additional atomic

formulasPi

(x

), meaning thatx

is labeled byPi

. If a labelPi

does not appear in a labeled graph G , then we alwaysinterpret Pi(G) as the empty set.

In monadic second-order formulas, we have additionalatomic formulas X(x) for set variables X and individual

variables x, and we admit existential and universalquantification over set variables. Set variables areinterpreted by sets of vertices, and the atomic formulaX(x) means that the vertex x is contained in the set X .



Monadic second-order logic (MSOL) is anextension of first-order logic that includes vertexand edge sets and quantification (universal and

existential) over these sets.

First-order logic is limited in its expressiveness: itcan only be used for local properties.

Monadic second-order logic’s set variables andset quantification allow for larger, more complexexpressions.



An example of a first-order formula φ is∀x ∃y [(edg(x, y)∨edg(y, x))∧ ￢(x =y)]

which is read “for every x there exists y such thatthe graph in question contains an edge from x to

y or the graph in question contains an edge fromy to x and it is not true that x equals y” or simplythe graph contains no isolated vertices.



Construction of branch decompositions

Tree Building: To construct a branchdecomposition, start with a partial branchdecomposition and refine this decompositionuntil the tree is ternary.

Note that separations are vital to the constructionof a branch decomposition because finding

separations will help refine partial branchdecompositions into branch decompositions.



The One Split

Given a partial branch decomposition, the studiedrefinements are one splits and two splits. Let G representour input graph and let (T1, ν) be a partial branchdecomposition of G. Let v be a non-leaf node of T1 with

degree greater than three and denote Dv as the set ofedges incident with v. For a set S ⊆ V (G), let he(S) denotea hyperedge where the ends of the hyperedge are theelements in S. Define Hv as the hypergraph constructedfrom the union of hyperedges he(mid(e)) for all e ∈Dv.

Let (X,Y ) be a separation of Hv. Create the tree T2 byreplacing v with nodes x and y and the edge xy where xwould be incident with the edges that correspond to E(X)and y would be incident with the edges that correspond toE(Y ).



The Two Split

Let G, (T1, ν), and v be defined as previous. Let ebe an edge incident with v and let he(e) denotethe hyperedge of Hv that corresponds to e.

Let (X,Y ) be a separation of the hypergraph Hv \{he(e)}. W.L.O.G assume that the cardinality ofE(X) is at most the cardinality of E(Y ). If thecardinality of E(X) is greater than one, create T2by adding new nodes x and y and edges vx andvy to T1 with x incident with the edgescorresponding to E(X) and y incident with theedges corresponding to E(Y ).



O.w, create T2 by inserting a new node y andedge vy with y incident with the edges

corresponding to E(Y ).

In order to build a branch decomposition, startwith a partial branch decomposition whose treeis a star and conduct a sequence of one and twosplits to achieve a branch decomposition.



Some definitions

A partial branch decomposition (T, ν) of a graphG is called extendible given that β(Hv)≤ β(G) forevery non-leaf node v ∈ V (T).

A separation is called greedy or safe if the nextpartial branch decomposition created by the useof the separation in conjunction with a one or

two split is extendible if the previous partialbranch decomposition was extendible.



Other types of safe separations:2-separations and 3-separations

First, given a partial branch decomposition of abiconnected graph, if a separation (X,Y ) is found suchthat |V (X) ∩ V (Y )| = 2 then (X,Y ) is safe.

All 3-separations (X,Y ) are safe unless V (X) ∩ V (Y )corresponds to an independent set in G and eitherV (X)\V (Y ) or V (Y )\V (X) has cardinality one.



Branch Decompositions and algorithms

Planar Graphs:

For planar (hyper )graphs, there exists apolynomial time algorithm called the ratcatchermethod to compute the branch width.

Let G be a graph with node set V (G) and edge setE(G). Let T be a tree having |V (G)| leaves inwhich every non-leaf node has degree three. Letμ be a bijection between the nodes of G and theleaves of T . The pair (T , μ) is called a carvingdecomposition of G.



Notice that removing an edge e of T partitions the nodesof G into two subsets Ae and Be.

The cut set of e is the set of edges that are incident with

nodes in both Ae and Be (also denoted δ(Ae) or δ(Be)). The width of a carving decomposition (T , μ) is the

maximum cardinality of the cut sets for all edges in T .

The carving width for G, κ(G), is the minimum width overall carving decompositions of G.

The ratcatcher method is really an algorithm to computethe carving width for planar graphs.



General Graphs

For general graphs, most work has been doneutilizing heuristics to actually construct branchdecompositions. Cook and Seymour gave a

heuristic algorithm to produce branchdecompositions. Their heuristic is based onspectral graph theory and the work of Alon.Moreover, Hicks also found another branchwidth heuristic that was comparable to the

algorithm of Cook and Seymour. This heuristicfinds separations by minimal vertex separatorsbetween diameter pairs.



Branch Decomposition Based Algorithms

The blueprint of a branch decomposition basedalgorithm typically consists of two steps:

transforming the tree of the branchdecomposition of a graph G into a rooted binarytree;

and visiting the nodes of the tree in post-depth-

first search order in order to generate a solutionfor the problem of interest.



The tree T is transformed into a rooted binary tree byselecting an edge ab and replacing ab by a node r, theroot, and edges ar and rb. Also, every node v of T that is

not a leaf node and not the root of the tree is adjacent tothree nodes: a parent node p; a left child lc; and a rightchild rc.

The algorithm then visits the tree nodes in post-depth-first order and builds a set of partial solutions for each

tree node. By visiting the tree in post-depth-first order, all

descendants of tree node v are visited before v is visited.



A branch decomposition based algorithm wasoffered by Cook and Seymour for the ring-routing-problem, which arises in the design of

reliable cost effective synchronous opticalnetworks (SONET networks) and wasincorporated into commercial software forTelcordia Technologies (formerly Bellcore).



Hicks has also developed computationally efficient branchdecomposition based algorithms for minor containmentand optimal branch decompositions.

One is also referred to the work of Christian for otherpractical algorithms.

Some examples of branch decomposition basedalgorithms proposed in theory are the notable work ofFomin and Thilikos and the work of Alekhnovich and

Razborov who used the branch width of hypergraphs todesign a branch decomposition based algorithm in theoryto solve satisfiability problems.

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